Theorem joinimandres-P4.8b 400

Description: Join Two Implications Through Conjunction (Restrictive). †

Hypothesis

Ref	Expression
joinimandres-P4.8b.1	⊢ (𝛾 → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜗)))

Assertion

Ref	Expression
joinimandres-P4.8b	⊢ (𝛾 → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜗)))

Proof of Theorem joinimandres-P4.8b

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → (𝜑 ∧ 𝜒))
2	1	ndander-P3.9 174	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → 𝜑)
3		joinimandres-P4.8b.1	. . . . . 6 ⊢ (𝛾 → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜗)))
4	3	rcp-NDIMP1add1 208	. . . . 5 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜗)))
5	4	ndander-P3.9 174	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → (𝜑 → 𝜓))
6	2, 5	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → 𝜓)
7	1	ndandel-P3.8 173	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → 𝜒)
8	4	ndandel-P3.8 173	. . . 4 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → (𝜒 → 𝜗))
9	7, 8	ndime-P3.6 171	. . 3 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → 𝜗)
10	6, 9	ndandi-P3.7 172	. 2 ⊢ ((𝛾 ∧ (𝜑 ∧ 𝜒)) → (𝜓 ∧ 𝜗))
11	10	rcp-NDIMI2 224	1 ⊢ (𝛾 → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜗)))

Syntax hints:   → wff-imp 10   ∧ wff-and 132

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133

This theorem is referenced by:  joinimandres-P4.8b.RC  401  joinimandres-P4.8b.CL  402  joinimandres2-P4  580  joinimandres3-P4  582